2 Answers
2

俺が何をしたっていうんだ。
What the heck did I do? (implies "I did nothing wrong.")

一人だから、何だというんだ。
So what if I am alone? (implies "There is nothing wrong even if I'm alone.")

彼女の何がダメだっていうの。
What the heck is wrong with her? (implies "There is nothing wrong with her.")

In your examples, ～っていうんだ, ～というんだ, ～っていうの are semantically roughly the same. They form a rhetoric question that expects "nothing" as an implied answer. って is a colloquial variation of quotative-と. The の/ん near the end is for seeking clarification. いう in this context is 言う, but is like "to mean" or "to think" rather than "to say out loud".

Even though they share the same meaning, they sound differently depending on how you end a sentence.

I don't think there's any one way to translate it. But it's said in response to someone else.

俺が何をしたっていうんだ = What are you saying I did?/What are you accusing me of?"

一人だから、何だというんだ = Because I'm alone, what of it? What does it matter to you if I'm alone?"

彼女の何がダメだっていうの = "What about her is bad, you're saying?"

That probably didn't help. But it is like emphasis, and has maybe an accusatory tone to it? It's easy to see the emphasis in the second sentence, 何だというんだ (also 何だってんだ) is almost a set phrase, which I tried to exemplify in my translation with "what of it"? The first sentence has the speaker (probably) thinking they're being judged and they asked the other person, "what do you think I did?" And the third sentence sounds similar, with an accusatory tone, like maybe someone is making fun of the girl, or doesn't like her.